Steps:

• Use the Pythagorean Theorem to solve for cosθ
• Find the quadrant restriction to see if it is positive or negative
• Use Tangent Identity to find tanθ

Real World Application:

Trigonometric Identities are used to solve for angles of everyday objects. For example, we are trying to solve for the angle of the leaning tower. We can take the one angle that we know, which is the angle that we are looking to see the top of the building. We can use the Trigonometric identities to take that one angle that we know, then solve for the angle of the leaning tower. If you look up at a leaning building, and the sine of the angle you are looking at is 2/3, then you can use the Pythagorean to solve for sine (√ 5/2), then for tan to get -2√ 5/5.

• The steps that you use to find these values are the same as the example above.

SAT Prep Question:

If sinβ=3/5 and 0<β<π2, evaluate cos2β.

• The equation we use is 2cosβ = 1 - 2sinβ.
• Plug in the given values, then solve for cosβ.
• The answer will be 7/25

Steps:

• The first thing to do is to change everything to sine and cosine form, so we change csc²x to 1/sin² x.
• We can then bring the denominator to the numerator to change it to (1-sin² x)(sin² x)/1-sin² x.
• The two 1-sin² x cancel out and that just leaves us with sin² x

Real World Application:

When carpenters measure a triangular piece of wood and come up with a precise measurement of 1-sin²x/csc²x-1, they need to simplify it down to an easier number to measure out.

SAT Prep Question:

(sinx1/1 + cosx) + (1+cosx/sinx)

• Find the common denominator
• Multiply the numerators and leave the denominators factored
• Add the numerators
• Use the Pythagorean Identity
• Combine like terms, then factor the numerator
• Reduce, then use the reciprocal identity to get 2cscx